I hope this qualifies within the some room for nonsense rule. I am writing a story about aliens (related to the game mentioned in the other thread), and I wanted to give them an alien form of mathematics, which human researchers can barely understand or interpret.

Is this possible and plausible? I'll mention a few ideas I had (pseudo science is allowed, for the purpose of debunking), and hope someone might be kind enough to mention any that come to mind.

No Irrational Numbers:

Spoiler:

One idea I considered, was if the aliens lacked irrational numbers, and perhaps their number system revolved around PI. Of course, this implies that when dealing with whole numbers they'd get irrational numbers, doesn't it? The idea of something as horrifying to a mathematician as, "these people are beyond irrational numbers!" seems interesting to me, but perhaps it shouldn't be.

Exponential Numbers:

Spoiler:

"1" = 2. "2" = 2. "3" = 4. "4" = 8, etc. A system where each number is exponential could be interesting, and just about all calculations are done in curves and powers (if you're travelling the galaxy, no one said it was simple).

I'm naive as to how difficult it would be learn this system in an alien language. It will be tricky when you aren't sure what numbers J, Q, L and X represent, or even if they are numbers. After the way the math works is discovered, I imagine it won't be too hard to calculate the aliens' figures?

Psychic Images:

Spoiler:

What if, instead of seeing a symbol like 4, you just saw four objects? Or instead of seeing 2.5387777778, you just saw a partial-circle with that exact fraction? If it were possible for a star people to be able to just see the objects instead of representing them with numbers, this could be very difficult to interpret for Earth mathematicians. Of course, it's questionable whether this makes sense, and if the aliens can really tell the difference between 3.14159265359, and 3.14159265358.

Lovecraftian Mathematics:

Spoiler:

Just for fun, I'll mention this idea. Sometimes when you perform the math problem, it comes out exactly how you want it to. Other times, it doesn't, despite using the same formula. The only difference in whether it comes out right or not, whether the math succeeds or fails... is whether you think about it. It gets stranger, when you get to the geometry of non-Euclidean shapes.

There is already a number system with no irrationals. We usually denote it as Q, the set of rational numbers. I don't know what it would mean to have a number system that 1) is larger than the rationals and 2) contains no irrationals; what are the extra numbers, then? In any case, if you don't have irrationals, you definitely don't have pi.

You can have other number systems with all kinds of goofy properties though. In the p-adics, "integers" can be irrational, which is pretty trippy.

"Exponential numbers" is essentially writing in base 2, and would be fairly unsurprising to human scientists.

Psychic images sound cool. It would suggest that the aliens think and perceive in a totally different way than humans do, allowing them to make incredibly fine distinctions as easily as we read text.

No, even in theory, you cannot build a rocket more massive than the visible universe.

Really, any alien representation of math we know about will just take work to translate, but then can be dealt with. "They do arithmetic / counting / mechanics in a super weird way" wouldn't last.

Maybe one option would be to just make all of the relevant mathematics for what they are doing with interestellar travel so trivial to them that they don't know what steps they are following anymore. Like, I cannot explain to you how reading works. I just do it now. I would have a hard time teaching someone to read, other than, just be doing it. Or, when I was teaching Calc 2, I had a really hard time explaining fractions to my students. They were supposed to know! I was not supposed to have to teach this! I could draw the pizza pies on the blackboard and when that explanation didn't work for the kiddies, I just had to hang my head and send them elsewhere, I cannot explain 3rd grade math so well anymore.

Our scientists try to have a talk with the aliens, and they're a little frustrated... "We can try to get someone who specializes in early childhood education on the next ship... we just brought scientists the first time around, we were hoping... no no you're good people, you're cute."

LE4dGOLEM: What's a Doug?Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

If you want aliens who think in a fundamentally different way than we do, think about what the tortoise said to Achilles. The notion of direct inference is so fundamental to the way our minds work that it can't be explained without making use of the very concept you're trying to explain. "If you know A, and you know 'if A then B', then you know B" is exactly the type of if-then statement it's trying to explain.

You could suppose the aliens have other notions that are similarly fundamental to their thinking and cannot be explained. To us, a lot of their math would look like completely unjustified leaps of logic, but they're actually quite rigorous if only we could understand the alien concept. And every time we struggle through to an argument based on our own methods of logic it turns out the aliens were right all along, but got there much more easily thanks to the mysterious extra concept they have available to them.

May I suggest you read Stories of Your Life, a short story by Ted Chiang? It's a first contact story, and this is one of the elements discussed in it. Not exactly the same, but definitely a similar vibe.

Giant CalculatorThe computers the aliens use aren't used quite the same way. They compute about half of the problems the ship faces--the rest being calculated mentally by the engineers. The engineers need to manually input a lot of the answers, because the machines can't work them out as quickly.

There are some mathematical problems a human does seem better able to solve than a computer, so a technologically enhanced alien may be better still when dealing with incredibly complex ideas. In submarines for example many calculations are still done by hand. If much of the information was missing from the computers, and locked deep in the brain of some alien, it may be impossible to understand what's going on.

Does this seem a reasonable idea?

arbiteroftruth wrote:

Spoiler:

If you want aliens who think in a fundamentally different way than we do, think about what the tortoise said to Achilles. The notion of direct inference is so fundamental to the way our minds work that it can't be explained without making use of the very concept you're trying to explain. "If you know A, and you know 'if A then B', then you know B" is exactly the type of if-then statement it's trying to explain.

You could suppose the aliens have other notions that are similarly fundamental to their thinking and cannot be explained. To us, a lot of their math would look like completely unjustified leaps of logic, but they're actually quite rigorous if only we could understand the alien concept. And every time we struggle through to an argument based on our own methods of logic it turns out the aliens were right all along, but got there much more easily thanks to the mysterious extra concept they have available to them.

Spoiler:

Oh, I like this idea! It would be confusing enough with symbols that represent complex formulas that don't seem to be described. But it'd be stranger still if, "Hey! This doesn't add up!""That's because you haven't multiplied it by the speed of light.""But it doesn't say to do that.""Does it need to? We're dealing with hyper-speed travel, of course you need to add the speed of light when considering drag forces, as that's the maximum velocity a physical object outside the stasis field can apply. You'll also need to add the Yelk equation.""What is the Yelk equation?""My friend! If you can remember how to calculate the speed of light, then you remember the Yelk equation!"

When dealing with aliens who basically have a perfect ability to remember mathematical formula, and perhaps a programming that causes them to associate certain formulas with certain tasks and problems, you could shorthand a whole lot of stuff.

The only problem is that computers generally can't short-hand things, can they? They can't make logical leaps as humans can? Generally, you'd expect to be able to look at the computer's code and calculations in depth, and find out everything it does to come to a conclusion. Unless the computer is so advanced no one can work out how to get such a debugging window.

Meteoric wrote:

Spoiler:

There is already a number system with no irrationals. We usually denote it as Q, the set of rational numbers. I don't know what it would mean to have a number system that 1) is larger than the rationals and 2) contains no irrationals; what are the extra numbers, then? In any case, if you don't have irrationals, you definitely don't have pi.

You can have other number systems with all kinds of goofy properties though. In the p-adics, "integers" can be irrational, which is pretty trippy.

"Exponential numbers" is essentially writing in base 2, and would be fairly unsurprising to human scientists.

Psychic images sound cool. It would suggest that the aliens think and perceive in a totally different way than humans do, allowing them to make incredibly fine distinctions as easily as we read text.

Hmm, this gives me an idea.

Irrational MathA system where every number is irrational and calculated to the millionth or even the quadrillionth digit could be very interesting. The computer mightn't like it if you miss out on digits, due to the extreme accuracy necessary for hyper-speed travel. This also makes me wonder if it would be possible to use an irrational base (calculated to the quadrillionth digit making it accurate enough).

The question then being how the computer stores these numbers, as if they're just on the keyboard... you can just type out 2 + 2 without worrying about the quadrillion digits of the irrational. The aliens surely can't be expected to type out a quadrillion digits or the like, unless they do it mentally and their brains just work that precisely and quickly with the computers.

With this system, it'd surely take researchers a long time to use the computer, as PI has only been calculated to the trillionth digit, and this computer requires you to calculate for every number, all of which are irrational.

And thanks, I hoped someone might find that last idea interesting. It may be the one I go with, as the other ideas have questionable aspects.

doogly wrote:

Spoiler:

Really, any alien representation of math we know about will just take work to translate, but then can be dealt with. "They do arithmetic / counting / mechanics in a super weird way" wouldn't last.

Maybe one option would be to just make all of the relevant mathematics for what they are doing with interestellar travel so trivial to them that they don't know what steps they are following anymore. Like, I cannot explain to you how reading works. I just do it now. I would have a hard time teaching someone to read, other than, just be doing it. Or, when I was teaching Calc 2, I had a really hard time explaining fractions to my students. They were supposed to know! I was not supposed to have to teach this! I could draw the pizza pies on the blackboard and when that explanation didn't work for the kiddies, I just had to hang my head and send them elsewhere, I cannot explain 3rd grade math so well anymore.

Our scientists try to have a talk with the aliens, and they're a little frustrated... "We can try to get someone who specializes in early childhood education on the next ship... we just brought scientists the first time around, we were hoping... no no you're good people, you're cute."

Spoiler:

Yeah, that's what makes this challenging to try and present a bizarre form of mathematics. It may be impossible, or may just be impossible for humans.

Like a kid with a calculator who has forgotten how to calculate by hand. I like that idea. And... wow, the kids just didn't know fractions? Talk about a school system failure (no offence to you). I'm totally stealing that line, just so you know.

I like the idea of aliens with calculators who don't understand it. You just need to make sure the scientists can't take apart the calculator so as to understand it. Maybe the calculator is attached to a large brain, or maybe it's some kind of Tesla-looking electric calculator that gets results through voltage and energy (where it's just going to take a long time to safely test the voltage, amps, and resistance involved, and reverse-engineer how the aliens use these to math).

Zohar wrote:May I suggest you read Stories of Your Life, a short story by Ted Chiang? It's a first contact story, and this is one of the elements discussed in it. Not exactly the same, but definitely a similar vibe.

I will look that one up. Thank you Zohar, it may well give me some ideas on how to handle this. How would you rate its presentation for alien math (or whatever the subject was)?

Yes, I was just about to mention Story of Your Life. That's the one "Arrival" (the recent movie) is based off of. I'm not sure if I would say the story is necessarily better than the movie or vice-versa; they're different approaches to similar subject matter. But the movie definitely doesn't bother with the nifty mathy bits.

I wonder how far you could get disregarding precision? Like, an alien species who regards it as not merely eccentric, but unthinkably insane to start rattling off the digits of pi even past ten decimal places. One that makes do with 22/7 the majority of the time, and regards resorting to 355/113 as downright transgressive? After all, how many cases are there that you actually need particular precision with pi?

Also along those lines, I might envision a species who regards it as anathema to write "3.14159..." and insists that the series representation is the only reasonable thing to use.

Another thing that comes to mind is Sawyer's Calculating God, which posits a species that can intuitively perceive numbers as high as 46 (or something) in the same way humans can perceive groups of five or six objects without actually having to mentally enumerate them. Said species, however, actually has no representation for numbers larger than 46.

What if the universe they come from (or the bit of ours, because of *mumble mumble cough*) had a pi that was 22/7? Or 3. Or 4. They'd never have to have to deal with irrationals. (Yeah, we got to do something even more skewy to rid ourselves of root(2), etc, as well.. Or else they "refuse to deal with them in normal circumstances" like we avoid dealing with negative, or even imaginary, roots.)

But we're talking weirdness beyond mere "taking a different perspective on life" a la the Jatravartid delay in developing the wheel, in favour of the under-arm deodorant. Probably have to bring their own bubble-universes with them to stop their structures/molecules/atoms/nucleons/etc not being quite right.

I'm not sure we won't just be encountering aliens with different ideas of number base (or place-variable-bases, perhaps, with triangular numbers being useful, even a Sierpinskiesque manner of holding increasing values without running out of names) and some weird tricks we haven't thought of while they've ignored some of the ones we take for granted (like finding a different form of trigonometry that still gives the same answers when properly applied).

Although, thinking about it only as I write, there's always a chance that they have adopted a set-theory style of notation (in different symbolics) from the assumed similar origins of numbers to ours of "packaging up" tokens into administrative enclosures, then going a different way from us.

This is interesting - I've often wondered if the "real" numbers really were "real" in the sense that we make them real.

If the universe is "discrete" and there is nothing other than the naturals outside of our minds, then I could see a hyper-advanced civilization that doesn't believe in anything other than the naturals.

Maybe they don't actually see any utility in the ratio of the circumference to diameter - I mean, sure, they may recognize that it converges, and their mathematicians may joke about the possibilities of a civilization that places great stock in this convergence, but the truth could be be that after 10 or 11 decimal places you've basically handled everything, and after 20 or 30 you've covered every POSSIBLE position. Maybe they simply don't believe in a "perfect circle."

It turns out the punchline to Godel's second incompletness theorem is that all "consistent formalized system which contains elementary arithmetic" are inconsistent.

All of our sufficiently powerful formal mathematics is someone playing with useless toys that by happenstance sometimes can be used to predict things.

They use a form of logic that isn't related to ours. They cannot translate it to our system, because their system is consistent, while ours is not.

It can be used to model reality and generate predictions that are consistent with reality.

Our logical systems, proofs and arguments, to them, are like a 4 year old working out why the milk she drinks is cold: "snow is cold, snow is white, milk is white so milk I drink is cold". All of them. They sometimes reach conclusions that are true, but the system itself isn't fundamentally sound.

The conclusion is true (yes, the milk you drink is indeed cold), but the logic used to reach it is basically nonsense.

When they point out flaws in our logical system or an argument, they have difficulty. Translating it to our logical system is hard. When they do, it seems to be a completely unrelated statement about unrelated things that is true ("So, you know how Pluto and Charon are in orbit with a period longer Earth's rotation? Well, your argument is unsound, clearly. Why? Because Pluto and Charon are in orbit with a period longer than Earth's rotation. What do you mean you don't see the connection? This is hard to explain."). To them, the source of this translation clearly points out a flaw in the logical chain we are doing. To demonstrate, they pull out some bit of super science; it depends on the flaw in our logical argument. "If Pluto and Charon where not in orbit witha period longer than Earth's rotation, then this wouldn't work. No, you moving the planets wouldn't make this not work; not just any Pluto and Charon. Those ones, and this Earth."

They can not use number. The universe doesn't really have number in it anyhow, it is a cludge to attempt to compress information about it. They can understand numbers with effort, but it takes work and doesn't seem to help solve problems. When they translate they tend to use "a few", "many", and the like seemingly vague nouns.

We could also throw out linear speech. Their communication constructs ("sentences") might require 3 or more dimensions to graph out. You could imagine playing music and motion on multiple channels over space and time.

To draw an analogy, there are 2 dimensional programming languages where the program is a grid, with execution pointers running all over the place and output being on a similar grid. Imagine a language where you communicate by constructing such a program and handing it to the other person. The meanig of the sentence is based on the actions of the instruction pointers and the output.

Now make it higher dimensional and have non-deterministic instruction pointers.

And pretend it is something that complex, but there is no "running steps", just connected components whose meaning is non-local.

Then this isn't what they are communicating, but rather a simplified view of it, like how we might teach a dog to express its wants by touching symbols and train it to follow commands, compared to us writing a novel. Their everyday speech to each other is to that translation as a novel is to "sit".

One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

He could also be referencing the fact that any formal system containing Robinson arithmetic and a proof of its own consistency is inconsistent. So it's a good thing that ZFC can't prove its own consistency, because that would actually mean it was inconsistent.

Way I see it, if you are writing about aliens with a mathematical system incomprehensible to humans, you have about 4 options.1) Their system of mathematics can be completely different from ours, with different axioms as well as rules of inference, possibly with few or no real-world applications or applications to our flavor of mathematics we can see, but comprehensible to humans if they were to study it hard enough and translate it, which presumably hasn't happened yet in your story.2) Their system of mathematics requires much more brainpower and/or intelligence to comprehend than humans have, though it is not fundamentally different from our system of mathematics.3) The alien's system of math includes inference rules that are actually Turing-uncomputable, meaning it is extremely unlikely for humans or any extension of them to ever make sense of it or4) You can just BS your way through, say the math system is "incomprehensible" in whatever way you need for your story, and most readers will not question this because they do not have the mathematical background to start to think about what an alien system of mathematics would even be. With this approach, you can even have god-aliens that make no sense, like the ones in Contact that mess with mathematics itself by virtue of creating the universe, even though you would have to do a whole lot more than that to mess with mathematics itself.

Just for fun, I'll mention this idea. Sometimes when you perform the math problem, it comes out exactly how you want it to. Other times, it doesn't, despite using the same formula. The only difference in whether it comes out right or not, whether the math succeeds or fails... is whether you think about it. It gets stranger, when you get to the geometry of non-Euclidean shapes.

I don't think the geometry of non-Euclidean shapes is stranger than a math problem that comes out differently depending on "whether you think about it" Mathematicians already work in non-Euclidean geometry, but a math problem that doesn't always come out the same way? That would pretty much lead to the collapse of all mathematics, which would all be inconsistent.

Come to think of it, there's also a central point in David Brin's second Uplift trilogy, a universe in which humans are just so gosh-darn special in that pretty much every other galactic civilization has access to advanced technology early in its development. With advanced number-crunching techniques on hand, no one else has a particular need to develop calculus, because something like finding the area under a curve can just be brute-forced. But that means that in the end, the Earthlings are still capable of figuring out stuff that would ordinarily take hundreds or thousands of years even on advanced computer systems.